We now briefly review the numerical procedure for estimating the error threshold of a quantum memory.
Consider a family of codes of growing size, parameterized by \(d\) .
In the case of the RHG lattice, \(d\)  is the code distance (the weight of the minimal weight non-trivial logical operator) and the number of qubits is \(n = O( d^3 )\) .
Another parameter is the noise channel, which in our case is described by two numerical parameters: the noise variance \(\delta \)  and the swap-out probability \(p_0\) .
To estimate the error threshold, we run many trials of Monte Carlo simulations to determine the logical error rates as a function of our physical noise parameters.
This is done for different lattice sizes \(d\) .
In each trial, we generate homodyne measurement outcomes according to the noise parameters, then we run our error correction procedure (inner decoder followed by outer decoder as described in Sec:QEC), and finally check if error correction has been successful.
Let us assume that we fix \(p_0\)  and vary \(\delta \) .
Then if a threshold, \(\delta _c\) , exists, we expect to see the following behaviour.
For \(\delta > \delta _c\) , increasing the size of the code (increasing \(d\) ) increases the logical error rate.
But for \(\delta < \delta _c\) , increasing the size of the code exponentially decreases the logical error rate.
We note that the largest code sizes we consider involve \(n \approx 5000\)  qubits.
Simulation of such a large number of qubits is possible due to the fact that we use a classical noise channel to model approximate GKPs and the circuits we simulate belong to the Clifford group, which makes them efficiently classically simulable [1]}.
